A tuning fork of frequency 100 when sounded together with another tuning fork of unknown frequency produces 2 beats per second. On loading the tuning fork whose frequency is not known and sounded together with a tuning fork of frequency 100 produces one beat, then the frequency of the other tuning fork is

Options

A. 102
B. 98
C. 99
D. 101

Answer

To solve the problem, we need to analyze the information given about the two tuning forks and the beats produced when they are sounded together.

### Step-by-Step Solution:

1. **Understanding Beats**:
The number of beats produced when two tuning forks are sounded together is given by the absolute difference in their frequencies. If \( f_1 \) is the frequency of the known tuning fork (100 Hz) and \( f_2 \) is the frequency of the unknown tuning fork, then:
\[ |f_1 - f_2| = \text{Number of beats per second} \]

2. **First Scenario (2 beats per second)**:
When the unknown tuning fork is sounded with the 100 Hz fork, it produces 2 beats per second:
\[ |100 - f_2| = 2 \]
This gives us two possible equations:
\[ f_2 = 100 + 2 = 102 \quad \text{(1)} \]
or
\[ f_2 = 100 - 2 = 98 \quad \text{(2)} \]

3. **Second Scenario (1 beat per second)**:
When the unknown tuning fork is loaded, its frequency decreases. Now, it produces 1 beat per second with the 100 Hz fork:
\[ |100 - f_2'| = 1 \]
where \( f_2' \) is the frequency of the loaded tuning fork. This gives us:
\[ f_2' = 100 + 1 = 101 \quad \text{(3)} \]
or
\[ f_2' = 100 - 1 = 99 \quad \text{(4)} \]

4. **Considering the Effect of Loading**:
When the tuning fork is loaded, its frequency decreases. Therefore, we can conclude that:
\[ f_2' < f_2 \]
This means that the frequency after loading (from equations 3 and 4) must be less than the original frequency (from equations 1 and 2).

5. **Analyzing the Options**:
- If \( f_2 = 102 \) (from equation 1), then \( f_2' \) cannot be 101 or 99 because both are less than 102.
- If \( f_2 = 98 \) (from equation 2), then \( f_2' \) could be 99 (from equation 4), which is greater than 98, but this contradicts the fact that loading decreases the frequency.

6. **Conclusion**:
Since the only valid option that satisfies all conditions is:
\[ f_2 = 98 \text{ Hz} \]

### Final Answer:
The frequency of the other tuning fork is **98 Hz**.